How can I change transform?
I'm not sure if I got what you ask but it seems that your rectangular mesh outlined in green has its local axes not aligned to its vertical/horizontal ides (probably you applied its rotation for some reason), and now you wish to make your local axes "realigned" in order to be able to "move" the mesh along one of those axes.

If you need this, you have to help your "axes realignment" with a separate object, say an empty.


you add an empty
you place it on one of your mesh "corners" helping with snap tools
you also use snap tools to rotate the empty, aligning one of its axes to one of your mesh sides
then you copy the empty rotation value (eg: 138.516796°)
then you rotate the mesh of a value of "180° - 138.516796°"
now your mesh is perfectly aligned with the grid, so you apply this new mesh rotation, which gets you good "local" axes
then you rotate "back" your mesh of the same amount but inversed ("138.516796° - 180°" => note, in the animation below I did this wrong), to rotate back the mesh where it was at the beginning (but now you have aligned local axes)
then you can move your mesh along the desired direction


See how it works:



Hth